TFY4345: Computational Planetary Orbit Programme - The Runge-Kutta Algorithm - IT Assignment Help

Assignment Task:

The exercise considers the motion of planets in the solar system. Planetary motion is governed by  the Newtonian force law of gravity (see the Appendix for planetary motion, pdf slides). Notice that all planets in the solar system move approximately in the same plane, which meansthat two dimensional simulations will be sufficient. In the first two parts of the project (topics 1 and 2), the  motion of a single planet is investigated. For numerical values of the gravitational constant,  planetary masses, etc., please take a look at the Appendix. Students should work in pairs. Contact  the TAs in case you do not have a partner and would like to find one.  

Hand-over your assignment in BlackBoard (option under construction). You may provide your  report as a standard report (pdf) or as a pre-compiled Jupyter notebook file. Focus on the quality  of the figures that represent your results. There are no requirements for the text itself. 

Submission date: October 2nd, 2020 

1. Planetary motion. 

• First, design a planetary orbit programme based on the Runge-Kutta (RK) algorithm (4th order) which can reproduce the trajectory of Earth for fixed Sun starting from time min and ending at tmax using a time step τ (set by the user). The physical units of your programme should  be the astronomical units (AU, years and etc.). For Earth, it takes 1 year to complete a full revolution which provides us a (starting) velocity of 2π (AU/yr). Record the position (x- and y coordinates), velocity, kinetic energy, potential energy and total energy as functions of time. Include also the simpler Euler-Cromer (EC) algorithm in your code for comparison purposes. 

• It is important to perform a convergence test, and you should consider the effect of the time step τ on your trajectory. What would be good initial guesses for RK and EC? Why? For one particular planet (e.g., the Earth), optimize τ using the requirement of total energy conservation for both RK and EC. To this end, calculate the energy change over one orbit and plot it as a  function of τ. What type of behaviour do you expect? Choose an appropriate way of plotting the result. Compare between the Runge-Kutta and Euler-Cromer algorithm throughout. 

• Demonstrate Kepler's third law for all planets with nearly circular orbits (for parameters see the Appendix). Think about how to choose the initial conditions to obtain circular orbits. 

2. Perihelion of Mercury á la Einstein. 

We shall consider the stability of planetary orbits by considering the prediction by Einstein for the precession of the perihelion of Mercury based on the General Theory of Relativity (see the Appendix). The observed precession is of the order 566 arcseconds per century (0.1572 degrees), and it is mostly caused by the gravitational effect of other planets(as numerically calculated by early astronomers). However, there was still a peculiar contribution of 43 arcseconds per century which remained unexplained. Based on his ideas on gravitation, Einstein provided an analytical solution for this problem. 

• Start the simulation by considering the case without precession. Keep the Sun fixed and place  Mercury on an elliptical orbit by starting the simulation from perihelion (nearest point to the 

Sun) where the velocity is at its maximum. As before, record the position, velocity, kinetic energy, potential energy and total energy as functions of time, and make sure that you are using a proper time step. You may use the RK algorithm solely from now on. 

• The force law with Einstein’s correction states that the gravitational force becomes slightly modified from the Newtonian inverse-square form and leads to a slow precession (rotation) of the perihelion for repeated orbits. Modify your program to investigate this effect. The value of α = 1.1 × 10-8 AU2 derived by Einstein is very small. What can be done instead? Calculate for several larger values of α and use your results for estimating the correct value of precession.

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